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  1. ; ; (, Geometry & Topology)
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  2. ; ; ; (, Proceedings of the American Mathematical Society, Series B)
    We show a number of Toda brackets in the homotopy of the motivic bordism spectrum MGL and of the Real bordism spectrum MUR. These brackets are "red-shifting" in the sense that while the terms in the bracket will be of some chromatic height n, the bracket itself will be of chromatic height (n+1). Using these, we deduce a family of exotic multiplications in the π_{**}MGL-module structure of the motivic Morava K-theories, including non-trivial multiplications by 2. These in turn imply the analogous family of exotic multiplications in the π_{\star}MUR-module structure on the Real Morava K-theories. 
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  3. ; ; (, Advances in Mathematics)
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  4. ; ; ; (, Communications of the American Mathematical Society)
    In studying the “11/8-Conjecture” on the Geography Problem in 4-dimensional topology, Furuta proposed a question on the existence of Pin ⁡ ( 2 ) \operatorname {Pin}(2) -equivariant stable maps between certain representation spheres. A precise answer of Furuta’s problem was later conjectured by Jones. In this paper, we completely resolve Jones conjecture by analyzing the Pin ⁡ ( 2 ) \operatorname {Pin}(2) -equivariant Mahowald invariants. As a geometric application of our result, we prove a “10/8+4”-Theorem. We prove our theorem by analyzing maps between certain finite spectra arising from B Pin ⁡ ( 2 ) B\operatorname {Pin}(2) and various Thom spectra associated with it. To analyze these maps, we use the technique of cell diagrams, known results on the stable homotopy groups of spheres, and the j j -based Atiyah–Hirzebruch spectral sequence. 
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